Parkinson's men were given an early lead thanks to James Hanson's close range tap-in but two second half efforts for Gillingham - including one which appeared to come off Hanson - turned the game around for the Kent visitors.
Hanson had an opportunity to salvage a point for City deep in injury time but his close range header was superbly repelled by the reflexes of Gills 'keeper Stuart Nelson.
Speaking after the reversal, Parkinson wasn't afraid to admit that both he and his players need to share the blame for the defeat.